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Langevin Analysis for Time-Nonlocal Brownian Motion with Algebraic Memories and Delay Interactions

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Langevin Analysis for Time-Nonlocal Brownian Motion with Algebraic Memories and Delay Interactions Eur. Phys. J. B (2016) 89: 87 DOI: 10.1140/epjb/e2016-70079-5 Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions Matthew Chase, Tom J. McKetterick, Luca Giuggioli and V.M. Kenkre Eur. Phys. J. B (2016) 89: 87 DOI: 10.1140/epjb/e2016-70079-5 THE EUROPEAN PHYSICAL JOURNAL B Regular Article Langevin analysis for time-nonlocal Brownian motion with algebraic memories and delay interactions Matthew Chase1,a, Tom J. McKetterick2,3, Luca Giuggioli2,3,4, and V.M. Kenkre1 1 Consortium of the Americas for Interdisciplinary Science and the Department of Physics and Astronomy, University of New Mexico, Albuquerque − New Mexico 87131, USA 2 Bristol Centre for Complexity Sciences, University of Bristol, BS2 8BB, Bristol, UK 3 Department of Engineering Mathematics, University of Bristol, BS8 1UB, Bristol, UK 4 School of Biological Sciences, University of Bristol, BS8 1TQ, Bristol, UK Received 2 February 2016 / Received in final form 13 February 2016 Published online 6 April 2016 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2016 Abstract. Starting from a Langevin equation with memory describing the attraction of a particle to a center, we investigate its transport and response properties corresponding to two special forms of the memory: one is algebraic, i.e., power-law, and the other involves a delay. We examine the properties of the Green function of the Langevin equation and encounter Mittag-Leffler and Lambert W-functions well-known in the literature. In the presence of white noise, we study two experimental situations, one involving the motional narrowing of spectral lines and the other the steady-state size of the particle under consideration. By comparing the results to counterparts for a simple exponential memory, we uncover instructive similarities and differences. Perhaps surprisingly, we find that the Balescu-Swenson theorem that states that non-Markoffian equations do not add anything new to the description of steady-state or equilibrium observables is violated for our system in that the saturation size of the particle in the steady- state depends on the memory function utilized. A natural generalization of the Smoluchowski equation for the time-local case is examined and found to satisfy the Balescu-Swenson theorem and describe accurately the first moment but not the second and higher moments. We also calculate two-time correlation functions for all three cases of the memory, and show how they differ from (tend to) their Markoffian counterparts at small (large) values of the difference between the two times. 1 Introduction The word “essentially” refers here to the fact that ini- tial conditions dictate in this context an extra term not The time evolution of the probability density of a parti- present in the diffusion equation. The term decays at the cle attracted to a center via harmonic forces while being rate γ in a well-known way. simultaneously subjected to white Gaussian noise is of in- Our interest in the present paper is in systems in terest in a large variety of contexts. The label Ornstein- which the attraction to the center proceeds via a time- Uhlenbeck is attached to the system or process under nonlocal process. Memory-possessing Langevin equations such situations and the governing equation is said to be have come under investigation in various unrelated con- the Smoluchowski equation. It is ubiquitous in statistical texts in the past, but it is appropriate to say that mechanics [1,2] and, in a one-dimensional system takes modern work on the topic appears to have begun with the form Budini and C´aceres [3]. In a report with far-reaching conclusions, they studied generalized Langevin equations ∂P(x, t) ∂ ∂P(x, t) producing many interesting results, but restricted their = γxP(x, t)+D (1) ∂t ∂x ∂x investigations primarily to the exponential memory, fo- cusing on various kinds of non-Gaussian noise includ- where γ measures the rate of attraction to the fixed cen- ing radioactive, Poisson, and Abel noise. They touched ter and D is the diffusion constant. Its solutions are well- upon algebraic memory using fractional derivatives but known as being essentially identical to those of the sim- assumed zero dissipation in their analysis, i.e., γ =0. ple diffusion equation (no attractive center) provided the In a more recent study, they analyzed stationary proper- time t itself undergoes a saturation transformation via ties of Langevin equations with memories of the algebraic the Ornstein-Uhlenbeck prescription: t → (1 − e−2γt)/2γ. and delay type [4]asdidDrozdov[5] using functionals to characterize various noise distributions. There is also a e-mail: [email protected] an intriguing report in the literature by Fiscina et al. [6] Page 2 of 15 Eur. Phys. J. B (2016) 89: 87 of related ideas to the behavior of vibrated granular ma- one-variable description but to introduce a memory func- terial: they found that the observed asymptotic spectral tion relating the time derivative of x to an appropriate density could be well described using a Langevin equation function of x. Such a situation has been discussed in a with fractional derivatives. Work exploring the dynamic recent review of the mathematics of animal motion by behavior of a non-Markoffian Langevin equation was re- two of the present authors [14]. The origin of the memory ported by Bolivar [7] for arbitrary Gaussian noise corre- function or of time-nonlocality would be, in this case, the lation functions with a focus on the differentiability of incorporation of inertial terms. A quite different source of the displacement. A path integral analysis of a Langevin time-nonlocality is finiteness of the speed of propagation equation with exponential memory has also appeared [8]. of signals that are related to the attraction process. Ex- Fractional derivatives have been used [9] to introduce what amples may be found in delay formalisms as in a study are in essence memory effects into fractional probability of Alzheimer walks [15] and a recent analysis of pairwise density equations. General expositions of the subject that movement coordination [16] applicable, e.g., to a system have been highly useful for a number of years are available of foraging bats [17]. in various articles [10–13]. Although we will not use Fokker-Planck equations As is well-known, stochastic analysis may be under- for our analysis, we will begin our considerations with taken either via Langevin equations or via what has often the Smoluchowski equation (1). We do this because it been called a formalism based on Fokker-Planck equa- is easy to introduce our task in that manner and also tions. The former are ordinary differential equations for because there has been a lot of recent activity on that stochastic variables. The latter are partial differential equation. Thus, an analysis has shown [18]howthe equations for the deterministic probability density of those Smoluchowski equation may be applied to trapping situa- variables, an average of the stochastic variables carried out tions and a recent application to the spread of epidemics with the help of the probability density leading to expec- has been made [19] leading to an interesting description tation values. In the present paper, except for a discussion of the transmission of infection in diseases such as the leading into the problem as well as an interesting obser- Hantavirus [20]. vation at the end of the analysis, we will not use Fokker- Planck equations. A number of subtleties and controver- The rest of the paper is laid out as follows. We first sies appear in the discussion of Fokker-Planck equations show in the next section why an intuitively natural mem- and we plan to address them in a separate publication. ory generalization of an equation such as equation (1) Our focus in the present paper will be entirely on the does not work. We place our full focus therefore on the Langevin approach: the ordinary differential equation for Langevin equation with memory. We are generally inter- the basic stochastic variable under consideration is solved ested in biological processes far from thermodynamic equi- explicitly for the given memory and noise, and the desired librium. The near-equilibrium requirement of fluctuation- function of the variable, for instance an arbitrary power, dissipation relations between the noise and the memory is averaged taking into account the properties of the noise. then need not apply. To facilitate calculations we take the Thus, our general interest is in treating systems in noise to be white. Although this restriction is not essen- tial, we limit ourselves to this case on one hand because of which the Langevin equation for the coordinate xi of the ith particle is its general usefulness in providing insights into a variety of biological systems [21] ranging from physiological [22,23] t to ecological scales [24], and on the other hand because of dxi(t) = −γ dt φ(t − t )xi(t )+ξi(t), (2) the already rich scenario that we uncover even when the dt 0 noise is not colored. In Sections 3 and 4 respectively, we where ξi(t) denotes the noise and φ(t) is the memory func- obtain explicit usable prescriptions, specifically for alge- tion. The case when the memory is a δ-function, i.e., when braic and delay-type memories in the Langevin equation. the Langevin equation is time-local, corresponds to equa- We find the Green function for the case of an algebraic tion (1). Our specific interest is in two forms of the mem- memory to be the Mittag-Leffler function. In the case of ory: algebraic, i.e., power law, and incorporating a delay, a single delayed delta function, it is associated with the as we will explain below.
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